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2026
csat__math 30
2025
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2024
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2023
csat__math 30
2022
csat__math 41
2021
csat__math-humanities 30 csat__math-science 29
2020
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2019
csat__math-humanities 30 csat__math-science 30
2018
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2017
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2016
csat__math-A 30 csat__math-B 30
2015
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2014
csat__math-A 30 csat__math-B 30
2013
csat__math-humanities 30 csat__math-science 30
2012
csat__math-humanities 29 csat__math-science 30
2011
csat__math-humanities 30 csat__math-science 36
2010
csat__math-humanities 27 csat__math-science 33
2009
csat__math-humanities 27 csat__math-science 23
2008
csat__math-humanities 29 csat__math-science 14
2007
csat__math-humanities 30 csat__math-science 17
2006
csat__math-humanities 30 csat__math-science 26
2005
csat__math-humanities 27 csat__math-science 32
2019 csat__math-science

30 maths questions

Q1 2 marks Vectors Introduction & 2D Magnitude of Vector Expression View
For two vectors $\vec { a } = ( 1 , - 2 ) , \vec { b } = ( - 1,4 )$, what is the sum of all components of the vector $\vec { a } + 2 \vec { b }$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q2 2 marks Vectors 3D & Lines Section Division and Coordinate Computation View
For two points $\mathrm { A } ( 2 , a , - 2 ) , \mathrm { B } ( 5 , - 2,1 )$ in coordinate space, when the point that divides segment AB internally in the ratio $2 : 1$ lies on the $x$-axis, what is the value of $a$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q3 2 marks Differentiating Transcendental Functions Limit involving transcendental functions View
What is the value of $\lim _ { x \rightarrow 0 } \frac { x ^ { 2 } + 5 x } { \ln ( 1 + 3 x ) }$? [2 points]
(1) $\frac { 7 } { 3 }$
(2) 2
(3) $\frac { 5 } { 3 }$
(4) $\frac { 4 } { 3 }$
(5) 1
Q4 3 marks Conditional Probability Direct Conditional Probability Computation from Definitions View
For two events $A , B$, $A$ and $B ^ { C }$ are mutually exclusive events, and $$\mathrm { P } ( A ) = \frac { 1 } { 3 } , \mathrm { P } \left( A ^ { C } \cap B \right) = \frac { 1 } { 6 }$$ What is the value of $\mathrm { P } ( B )$? (Here, $A ^ { C }$ is the complement of $A$.) [3 points]
(1) $\frac { 5 } { 12 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 7 } { 12 }$
(4) $\frac { 2 } { 3 }$
(5) $\frac { 3 } { 4 }$
Q5 3 marks Exponential Equations & Modelling Geometric Properties of Exponential/Logarithmic Curves View
When the graph of the function $y = 2 ^ { x } + 2$ is translated in the $x$-direction by $m$ units, and this graph is symmetric to the graph of the function $y = \log _ { 2 } 8 x$ translated in the $x$-direction by 2 units with respect to the line $y = x$, what is the value of the constant $m$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q6 3 marks Conic sections Focal Distance and Point-on-Conic Metric Computation View
For a point P on the parabola $y ^ { 2 } = 12 x$ with focus F, when $\overline { \mathrm { PF } } = 9$, what is the $x$-coordinate of point P? [3 points]
(1) 6
(2) $\frac { 13 } { 2 }$
(3) 7
(4) $\frac { 15 } { 2 }$
(5) 8
Q7 3 marks Implicit equations and differentiation Compute slope at a point via implicit differentiation (single-step) View
What is the slope of the tangent line to the curve $e ^ { x } - x e ^ { y } = y$ at the point $( 0,1 )$? [3 points]
(1) $3 - e$
(2) $2 - e$
(3) $1 - e$
(4) $- e$
(5) $- 1 - e$
Q8 3 marks Binomial Distribution Find Parameters from Moment Conditions View
When the random variable $X$ follows the binomial distribution $\mathrm { B } \left( n , \frac { 1 } { 2 } \right)$ and satisfies $\mathrm { E } \left( X ^ { 2 } \right) = \mathrm { V } ( X ) + 25$, what is the value of $n$? [3 points]
(1) 10
(2) 12
(3) 14
(4) 16
(5) 18
Q9 3 marks Composite & Inverse Functions Derivative of an Inverse Function View
When the inverse function of $f ( x ) = \frac { 1 } { 1 + e ^ { - x } }$ is $g ( x )$, what is the value of $g ^ { \prime } ( f ( - 1 ) )$? [3 points]
(1) $\frac { 1 } { ( 1 + e ) ^ { 2 } }$
(2) $\frac { e } { 1 + e }$
(3) $\left( \frac { 1 + e } { e } \right) ^ { 2 }$
(4) $\frac { e ^ { 2 } } { 1 + e }$
(5) $\frac { ( 1 + e ) ^ { 2 } } { e }$
Q10 3 marks Probability Definitions Probability Involving Algebraic or Number-Theoretic Conditions View
A bag contains 7 marbles, each labeled with a natural number from 2 to 8. When 2 marbles are drawn simultaneously from the bag, what is the probability that the two natural numbers on the drawn marbles are coprime? [3 points]
(1) $\frac { 8 } { 21 }$
(2) $\frac { 10 } { 21 }$
(3) $\frac { 4 } { 7 }$
(4) $\frac { 2 } { 3 }$
(5) $\frac { 16 } { 21 }$
Q11 3 marks Quadratic trigonometric equations View
When $0 \leq \theta < 2 \pi$, the range of all values of $\theta$ such that the quadratic equation in $x$ $$6 x ^ { 2 } + ( 4 \cos \theta ) x + \sin \theta = 0$$ has no real roots is $\alpha < \theta < \beta$. What is the value of $3 \alpha + \beta$? [3 points]
(1) $\frac { 5 } { 6 } \pi$
(2) $\pi$
(3) $\frac { 7 } { 6 } \pi$
(4) $\frac { 4 } { 3 } \pi$
(5) $\frac { 3 } { 2 } \pi$
Q12 3 marks Permutations & Arrangements Distribution of Objects into Bins/Groups View
The number of ways to distribute 8 identical chocolates to four students $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }$ according to the following rules is? [3 points] (가) Each student receives at least 1 chocolate. (나) Student A receives more chocolates than student B.
(1) 11
(2) 13
(3) 15
(4) 17
(5) 19
Q13 3 marks Vectors: Lines & Planes Find Intersection of a Line and a Plane View
In coordinate space, the $x$-coordinate of the point where the plane passing through the point $( 2,0,5 )$ and containing the line $x - 1 = 2 - y = \frac { z + 1 } { 2 }$ meets the $x$-axis is? [3 points]
(1) $\frac { 9 } { 2 }$
(2) 4
(3) $\frac { 7 } { 2 }$
(4) 3
(5) $\frac { 5 } { 2 }$
Q14 4 marks Exponential Equations & Modelling Exponential Inequality Solving View
When the graphs of the quadratic function $y = f ( x )$ and the linear function $y = g ( x )$ are as shown in the figure, the sum of all natural numbers $x$ satisfying the inequality $$\left( \frac { 1 } { 2 } \right) ^ { f ( x ) g ( x ) } \geq \left( \frac { 1 } { 8 } \right) ^ { g ( x ) }$$ is? [4 points] [Figure]
(1) 7
(2) 9
(3) 11
(4) 13
(5) 15
Q15 4 marks Normal Distribution Normal Distribution Combined with Total Probability or Bayes' Theorem View
The commute time of employees at a certain company on a certain day follows a normal distribution with mean 66.4 minutes and standard deviation 15 minutes. Among employees whose commute time is 73 minutes or more, 40\% used the subway, and among employees whose commute time is less than 73 minutes, 20\% used the subway, with the remaining employees using other transportation. What is the probability that a randomly selected employee from those who commuted on that day used the subway? (Here, when $Z$ is a random variable following the standard normal distribution, $\mathrm { P } ( 0 \leq Z \leq 0.44 ) = 0.17$.) [4 points]
(1) 0.306
(2) 0.296
(3) 0.286
(4) 0.276
(5) 0.266
Q16 4 marks Integration by Substitution Substitution Combined with Symmetry or Companion Integral View
A continuous function $f ( x )$ defined on $x > 0$ satisfies $$2 f ( x ) + \frac { 1 } { x ^ { 2 } } f \left( \frac { 1 } { x } \right) = \frac { 1 } { x } + \frac { 1 } { x ^ { 2 } }$$ for all positive $x$. What is the value of $\int _ { \frac { 1 } { 2 } } ^ { 2 } f ( x ) d x$? [4 points]
(1) $\frac { \ln 2 } { 3 } + \frac { 1 } { 2 }$
(2) $\frac { 2 \ln 2 } { 3 } + \frac { 1 } { 2 }$
(3) $\frac { \ln 2 } { 3 } + 1$
(4) $\frac { 2 \ln 2 } { 3 } + 1$
(5) $\frac { 2 \ln 2 } { 3 } + \frac { 3 } { 2 }$
Q17 4 marks Combinations & Selection Counting Functions or Mappings with Constraints View
The following is a process to find the number of functions $f$ such that the number of elements in the range of the composite function $f \circ f$ is 5, for the set $X = \{ 1,2,3,4,5,6 \}$ and the function $f : X \rightarrow X$.
Let the ranges of the function $f$ and the function $f \circ f$ be $A$ and $B$, respectively. If $n ( A ) = 6$, then $f$ is a bijection, and $f \circ f$ is also a bijection, so $n ( B ) = 6$. Also, if $n ( A ) \leq 4$, then $B \subset A$, so $n ( B ) \leq 4$. Therefore, we only need to consider the case where $n ( A ) = 5$, that is, $B = A$.
(i) The number of ways to choose a subset $A$ of $X$ with $n ( A ) = 5$ is (가).
(ii) For the set $A$ chosen in (i), let $k$ be the element of $X$ that does not belong to $A$. Since $n ( A ) = 5$, the number of ways to choose $f ( k )$ from the set $A$ is (나).
(iii) For $A = \left\{ a _ { 1 } , a _ { 2 } , a _ { 3 } , a _ { 4 } , a _ { 5 } \right\}$ chosen in (i) and $f ( k )$ chosen in (ii), since $f ( k ) \in A$ and $A = B$, we have $A = \left\{ f \left( a _ { 1 } \right) , f \left( a _ { 2 } \right) , f \left( a _ { 3 } \right) , f \left( a _ { 4 } \right) , f \left( a _ { 5 } \right) \right\} \cdots ( * )$. The number of cases satisfying (*) is equal to the number of bijections from set $A$ to set $A$, so $\square$ (다).
Therefore, by (i), (ii), and (iii), the number of functions $f$ we seek is $\square$ (가) $\times$ $\square$ (나) $\times$ $\square$ (다).
When the numbers that fit (가), (나), and (다) are $p , q , r$ respectively, what is the value of $p + q + r$? [4 points]
(1) 131
(2) 136
(3) 141
(4) 146
(5) 151
Q18 4 marks Radians, Arc Length and Sector Area View
As shown in the figure, in a right triangle ABC with $\overline { \mathrm { AB } } = 1 , \angle \mathrm { B } = \frac { \pi } { 2 }$, let D be the intersection of the angle bisector of $\angle \mathrm { C }$ and segment AB, and let E be the intersection of the circle with center A and radius $\overline { \mathrm { AD } }$ and segment AC. When $\angle \mathrm { A } = \theta$, let $S ( \theta )$ be the area of sector ADE and $T ( \theta )$ be the area of triangle BCE. What is the value of $\lim _ { \theta \rightarrow 0 + } \frac { \{ S ( \theta ) \} ^ { 2 } } { T ( \theta ) }$? [4 points] [Figure]
(1) $\frac { 1 } { 4 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 3 } { 4 }$
(4) 1
(5) $\frac { 5 } { 4 }$
Q19 4 marks Vectors: Lines & Planes Perpendicular/Orthogonal Projection onto a Plane View
For a tetrahedron ABCD with an equilateral triangle BCD of side length 12 as one face, let H be the foot of the perpendicular from vertex A to plane BCD. The point H lies inside triangle BCD. The area of triangle CDH is 3 times the area of triangle BCH, the area of triangle DBH is 2 times the area of triangle BCH, and $\overline { \mathrm { AH } } = 3$. Let M be the midpoint of segment BD, and let Q be the foot of the perpendicular from point A to segment CM. What is the length of segment AQ? [4 points]
(1) $\sqrt { 11 }$
(2) $2 \sqrt { 3 }$
(3) $\sqrt { 13 }$
(4) $\sqrt { 14 }$
(5) $\sqrt { 15 }$
Q20 4 marks Tangents, normals and gradients Find tangent line with a specified slope or from an external point View
From the point $\left( - \frac { \pi } { 2 } , 0 \right)$, tangent lines are drawn to the curve $y = \sin x ( x > 0 )$, and when the $x$-coordinates of the points of tangency are listed in increasing order, the $n$-th number is denoted as $a _ { n }$. For all natural numbers $n$, which of the following statements in the given options are correct? [4 points]
Options ㄱ. $\tan a _ { n } = a _ { n } + \frac { \pi } { 2 }$ ㄴ. $\tan a _ { n + 2 } - \tan a _ { n } > 2 \pi$ ㄷ. $a _ { n + 1 } + a _ { n + 2 } > a _ { n } + a _ { n + 3 }$
(1) ㄱ
(2) ㄱ, ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q21 4 marks Implicit equations and differentiation Functional equation with derivative conditions View
A function $f ( x )$ that is differentiable on the entire set of real numbers satisfies the following conditions. What is the value of $f ( - 1 )$? [4 points] (가) For all real numbers $x$, $$2 \{ f ( x ) \} ^ { 2 } f ^ { \prime } ( x ) = \{ f ( 2 x + 1 ) \} ^ { 2 } f ^ { \prime } ( 2 x + 1 ).$$ (나) $f \left( - \frac { 1 } { 8 } \right) = 1 , f ( 6 ) = 2$
(1) $\frac { \sqrt [ 3 ] { 3 } } { 6 }$
(2) $\frac { \sqrt [ 3 ] { 3 } } { 3 }$
(3) $\frac { \sqrt [ 3 ] { 3 } } { 2 }$
(4) $\frac { 2 \sqrt [ 3 ] { 3 } } { 3 }$
(5) $\frac { 5 \sqrt [ 3 ] { 3 } } { 6 }$
Q22 3 marks Permutations & Arrangements Factorial and Combinatorial Expression Simplification View
Find the value of ${}_{6}\mathrm{P}_{2} - {}_{6}\mathrm{C}_{2}$. [3 points]
Q23 3 marks Reciprocal Trig & Identities View
When $\tan \theta = 5$, find the value of $\sec ^ { 2 } \theta$. [3 points]
Q24 3 marks Variable acceleration (vectors) View
The position $( x , y )$ of a point P moving on the coordinate plane at time $t ( t \geq 0 )$ is $$x = 1 - \cos 4 t , y = \frac { 1 } { 4 } \sin 4 t.$$ When the speed of point P is maximum, find the magnitude of the acceleration of point P. [3 points]
Q25 3 marks Integration by Parts Definite Integral Evaluation by Parts View
Find the value of $\int _ { 0 } ^ { \pi } x \cos ( \pi - x ) d x$. [3 points]
Q26 4 marks Confidence intervals Algebraic problem using two confidence intervals View
The daily leisure activity time of residents in a certain region follows a normal distribution with mean $m$ minutes and standard deviation $\sigma$ minutes. When 16 residents are randomly sampled and the sample mean of daily leisure activity time is 75 minutes, the 95\% confidence interval for the population mean $m$ is $a \leq m \leq b$. When 16 residents are randomly sampled again and the sample mean of daily leisure activity time is 77 minutes, the 99\% confidence interval for the population mean $m$ is $c \leq m \leq d$. Find the value of $\sigma$ that satisfies $d - b = 3.86$. (Here, when $Z$ is a random variable following the standard normal distribution, $\mathrm { P } ( | Z | \leq 1.96 ) = 0.95 , \mathrm { P } ( | Z | \leq 2.58 ) = 0.99$.) [4 points]
Q27 4 marks Independent Events View
A die is rolled once. Let A be the event that an odd number appears, and for a natural number $m$ with $m \leq 6$, let B be the event that a divisor of $m$ appears. Find the sum of all values of $m$ such that the two events A and B are independent. [4 points]
Q28 4 marks Conic sections Optimization on Conics View
There is an ellipse $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 33 } = 1$ with foci $\mathrm { F } , \mathrm { F } ^ { \prime }$. For a point P on the circle $x ^ { 2 } + ( y - 3 ) ^ { 2 } = 4$, let Q be the point with positive $y$-coordinate among the points where the line $\mathrm { F } ^ { \prime } \mathrm { P }$ meets this ellipse. Find the maximum value of $\overline { \mathrm { PQ } } + \overline { \mathrm { FQ } }$. [4 points]
Q29 4 marks Vectors Introduction & 2D Area Computation Using Vectors View
In the coordinate plane, for a triangle ABC with area 9, let P, Q, R be points moving on the three sides AB, BC, CA respectively. When $$\overrightarrow { \mathrm { AX } } = \frac { 1 } { 4 } ( \overrightarrow { \mathrm { AP } } + \overrightarrow { \mathrm { AR } } ) + \frac { 1 } { 2 } \overrightarrow { \mathrm { AQ } }$$ is satisfied, the area of the region represented by point X is $\frac { q } { p }$. Find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]
Q30 4 marks Differentiating Transcendental Functions Compute derivative of transcendental function View
For a cubic function $f ( x )$ with leading coefficient $6 \pi$, the function $g ( x ) = \frac { 1 } { 2 + \sin ( f ( x ) ) }$ has a local maximum or minimum at $x = \alpha$, and when all $\alpha \geq 0$ are listed in increasing order as $\alpha _ { 1 }$, $\alpha _ { 2 } , \alpha _ { 3 } , \alpha _ { 4 } , \alpha _ { 5 } , \cdots$, the function $g ( x )$ satisfies the following conditions. (가) $\alpha _ { 1 } = 0$ and $g \left( \alpha _ { 1 } \right) = \frac { 2 } { 5 }$. (나) $\frac { 1 } { g \left( \alpha _ { 5 } \right) } = \frac { 1 } { g \left( \alpha _ { 2 } \right) } + \frac { 1 } { 2 }$ When $g ^ { \prime } \left( - \frac { 1 } { 2 } \right) = a \pi$, find the value of $a ^ { 2 }$. (Here, $0 < f ( 0 ) < \frac { \pi } { 2 }$.) [4 points]